"That we agree on", the USA official said.
Both officials spoke this week to The Associated Press on condition of anonymity in line with regulations.
The officials emphasize the discussions are ongoing.
They will join troops pushing into the Old City area from the south. Government forces launched a major offensive to recapture Mosul, the last major urban stronghold of Isis in Iraq, in October.
Many have been leaving the Musherfa district of western Mosul after us -backed Iraqi forces took it from Islamic State on Friday, hoping to reach the relative safety of the eastern banks of the river. Joseph Votel, head of US Central Command, said during a US congressional hearing in March that the offensive had left 774 Iraqi security forces dead and 4,600 wounded. After all, the last time USA troops pulled out of Iraq, they were back just three years later.
Mattis said the number of troops would be "several thousand... similar to what we have now, maybe a little more".
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The Pentagon has close to 7,000 USA troops in Iraq, many not publicly acknowledged because they are on temporary duty or under specific personnel rules.
So far, in its counterterrorism strategy, the Trump administration has continued the Obama administration's preference for relying on airstrikes, special forces raids, and training and equipping local forces, rather than large, costly, and politically unpopular troop deployments.
"[The source said] that these people were fleeing the clashes, the fighting [that is] going on between Iraqi security forces and IS in that neighborhood of western Mosul, and that they had taken refuge in a "school house" in that area", Aljazeera's Mohammed Jamjoon, reporting from Erbil in northern Iraq, said.
Such an agreement would underscore how the fight against IS has drawn the USA into a deepening role in Iraq. If the two nations agree on terms, Iraq would join Afghanistan in having USA troops stationed there for the foreseeable future.
A densely populated city with a lot of displaced civilians and United States and Iraqi airstrikes leveling a number of residences, it turned out the school was actually full of civilians, with at least 68 civilians being killed in the attack, including 47 women and children. The US-led global coalition is providing key air and ground support to the offensive on the Islamic State's de facto capital in Iraq.
"We're ferrying Syrian Democratic Forces across the river and bringing them in by air, [which] has made considerable progress in first isolating and now liberating Tabqah", he said. There, al-Baghdadi designated the terror group's caliphate after they took over nearly a third of Iraq.